COLLADA export doesn't preserve animation for all bones
I figured out the first problem. Based on suggestions from this BlenderArtists thread and this extremely helpful Wildfire Games article, I figured out that I needed to bake the animation with the 'visual keying' box selected. To access this, I selected the skeleton, then navigated to Bake Action (Pose > Animation > Bake Action...) and checked the 'Visual Keying', 'Clear Constraints', and 'Clear Parents' boxes. These options also take care of rigs that include inverse kinematics.

As the Wildfire Games article suggested, try to avoid unchecking 'inherit rotation' on any bone, as it will not export properly.

(As for the tiny bones problem, I have no idea what's going on there)